Question

X-2 (x+2)(x-5) ≥0. пж​

Answer 1

Ответ:

хє [ 7-√209/4; 7+√209/4]

Пошаговое объяснение:

х-2(х+2)(х-5)≥0

х+(2х-4)(х-5)≥0

х-2х²+10х-4х+20≥0

7х-2х²+20≥0

-2х²+7х+20≥0

-2х²+7х+20=0

$x1 = \frac{7 + \sqrt{209} }{4}$

$x2 = \frac{7 - \sqrt{209} }{4}$

$- 2(x - \frac{7 + \sqrt{209} }{4} )(x - \frac{7 - \sqrt{209} }{4} ) \geqslant 0$

$(x - \frac{7 + \sqrt{209} }{4} )(x - \frac{7 - \sqrt{209} }{4} ) \leqslant 0$

функция { х- 7+√209/4≤0

х- 7-√209/4≥0

функция { х- 7+√209/4≥0

х- 7-√209/4≤0

функция { х≤ 7+√209/4

х≥ 7-√209/4

функция { х≥ 7+√209/4

х≤ 7-√209/4

хє [ 7-√209/4; 7+√209/4]

Answer 2

To solve the inequality $x^2 - 2(x+2)(x-5) \geq 0$, you can follow these steps:

1. First, expand the expression: $x^2 - 2(x+2)(x-5) = x^2 - 2(x^2 - 3x - 10) = x^2 - 2x^2 + 6x + 20$.

2. Combine like terms: $x^2 - 2x^2 + 6x + 20 = -x^2 + 6x + 20$.

3. Rewrite the inequality as: $-x^2 + 6x + 20 \geq 0$.

4. Now, solve this quadratic inequality. To do this, find the critical points by setting the expression equal to zero and then analyze the sign of the expression in each interval created by these points.

   $-x^2 + 6x + 20 = 0$.

   First, find the critical points by factoring or using the quadratic formula:

   $-x^2 + 6x + 20 = 0$.

   You can factor it as $-(x-10)(x+2) = 0$, which gives $x = 10$ and $x = -2$ as critical points.

5. Now, create intervals on the number line based on these critical points: $(-∞, -2]$, $[-2, 10]$, and $[10, ∞)$.

6. Pick a test point from each interval and plug it into the inequality to determine the sign of the expression in that interval.

   • For the interval $(-∞, -2]$, you can pick $x = -3$: $-(-3)^2 + 6(-3) + 20 = 9 - 18 + 20 = 11$, which is positive.

   • For the interval $[-2, 10]$, you can pick $x = 0$: $-0^2 + 6(0) + 20 = 20$, which is positive.

   • For the interval $[10, ∞)$, you can pick $x = 11$: $-11^2 + 6(11) + 20 = -121 + 66 + 20 = -35$, which is negative.

7. Analyze the sign of the expression in each interval:

   • $(-∞, -2]$: Positive
   • $[-2, 10]$: Positive
   • $[10, ∞)$: Negative

8. Finally, combine the information to determine when $-x^2 + 6x + 20$ is greater than or equal to zero:

   • The inequality is satisfied when $x$ is in the intervals where the expression is non-negative, which are $(-∞, -2]$ and $[-2, 10]$.

So, the solution to the inequality $-x^2 + 6x + 20 \geq 0$ is $x \in (-∞, -2] \cup [-2, 10]$.